L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.5 + 0.866i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)10-s + (0.258 + 0.965i)11-s + (1 − i)13-s + (0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (0.707 − 0.707i)20-s + 22-s + (0.707 + 1.22i)23-s + (−0.707 − 1.22i)26-s + (0.866 − 0.5i)28-s + (−0.707 + 0.707i)29-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.5 + 0.866i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)10-s + (0.258 + 0.965i)11-s + (1 − i)13-s + (0.707 + 0.707i)14-s + (0.500 + 0.866i)16-s + (0.707 − 0.707i)20-s + 22-s + (0.707 + 1.22i)23-s + (−0.707 − 1.22i)26-s + (0.866 − 0.5i)28-s + (−0.707 + 0.707i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9498316252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9498316252\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31805453203817913721404139963, −9.501406707472411817228088035197, −8.818481862990443199099290504530, −7.73815043337038344902473299348, −6.66070107109294961532046501872, −5.77723220967464839080916167677, −4.86771540135358274983423582009, −3.44869801082743206655978634160, −3.07913044444080022296992620655, −1.73385282925096996140256921178,
0.916555695665794486249253442370, 3.28696684098421995474206715054, 4.20282713880007494650671736155, 4.81347010661278597834797307657, 6.19279785774573361250190923312, 6.51701473250701652869459753821, 7.70004316649583712807313050952, 8.486173433749881895052357199133, 9.013158482765203736858540063163, 9.875904473255781215201332386259